This is a jukugo word, which usually means on'yomi readings from the kanji. Since 大 has two on'yomi readings, here's a mnemonic to help you remember which one to use:
You went to a specific private college just to study dyes (だい). You're so into dyes that you decided to go to this extremely expensive private university that specializes in dyes. Good for you!
Private college tuition sure is expensive.
I'm teaching English at a private uni in Osaka.
As for the school festival at the private university you told me about the other day, could you count me out this time? I’m sorry, don’t get mad at me. I’ll contact you when I have free time.